85,008 research outputs found
Pattern forming pulled fronts: bounds and universal convergence
We analyze the dynamics of pattern forming fronts which propagate into an
unstable state, and whose dynamics is of the pulled type, so that their
asymptotic speed is equal to the linear spreading speed v^*. We discuss a
method that allows to derive bounds on the front velocity, and which hence can
be used to prove for, among others, the Swift-Hohenberg equation, the Extended
Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that
the dynamically relevant fronts are of the pulled type. In addition, we
generalize the derivation of the universal power law convergence of the
dynamics of uniformly translating pulled fronts to both coherent and incoherent
pattern forming fronts. The analysis is based on a matching analysis of the
dynamics in the leading edge of the front, to the behavior imposed by the
nonlinear region behind it. Numerical simulations of fronts in the
Swift-Hohenberg equation are in full accord with our analytical predictions.Comment: 27 pages, 9 figure
Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
Fronts that start from a local perturbation and propagate into a linearly
unstable state come in two classes: pulled and pushed. ``Pulled'' fronts are
``pulled along'' by the spreading of linear perturbations about the unstable
state, so their asymptotic speed equals the spreading speed of linear
perturbations of the unstable state. The central result of this paper is that
the velocity of pulled fronts converges universally for time like
. The parameters ,
, and are determined through a saddle point analysis from the
equation of motion linearized about the unstable invaded state. The interior of
the front is essentially slaved to the leading edge, and we derive a simple,
explicit and universal expression for its relaxation towards
. Our result, which can be viewed as a general center
manifold result for pulled front propagation, is derived in detail for the well
known nonlinear F-KPP diffusion equation, and extended to much more general
(sets of) equations (p.d.e.'s, difference equations, integro-differential
equations etc.). Our universal result for pulled fronts thus implies
independence (i) of the level curve which is used to track the front position,
(ii) of the precise nonlinearities, (iii) of the precise form of the linear
operators, and (iv) of the precise initial conditions. Our simulations confirm
all our analytical predictions in every detail. A consequence of the slow
algebraic relaxation is the breakdown of various perturbative schemes due to
the absence of adiabatic decoupling.Comment: 76 pages Latex, 15 figures, submitted to Physica D on March 31, 1999
-- revised version from February 25, 200
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Oscillatory Properties of Solutions of the Fourth Order Difference Equations with Quasidifferences
A class of fourth--order neutral type difference equations with
quasidifferences and deviating arguments is considered. Our approach is based
on studying the considered equation as a system of a four--dimensional
difference system. The sufficient conditions under which the considered
equation has no quickly oscillatory solutions are given
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